Moving magnet type planar motor control

ABSTRACT

A control system for a moving magnet type planar motor is disclosed that permits positioning with three degrees of freedom. Motor force ripple compensation is achievable with the control system, and cross coupling between translation forces and torque is substantially decreased.

FIELD OF THE INVENTION

[0001] The invention relates to planar motors. More particularly, theinvention is related to a control system for a moving magnet type planarmotor.

BACKGROUND OF THE INVENTION

[0002] Precision systems, such as those used in semiconductorprocessing, inspection and testing, often use linear motors forpositioning objects such as semiconductor wafers. Conventional precisionsystems include separate, stacked stages that permit movement alongperpendicular axes (i.e., an “X” stage stacked on a “Y” stage). Thesesystems typically are complex, heavy and inefficient in operation.Improved object positioning, particularly for use in lithographicinstruments, has been realized through the use of planar motors, whichadvantageously permit simplicity in design, weight savings, as well asenhanced precision and efficiency. Such a linear or planar motor, inprinciple, operates in accordance with the Lorentz law, which relatesthe force on a charged particle to its motion in an electromagneticfield. An object such as a stage in a lithography system may betranslated or propelled using the electromagnetic force generated by awire or coil carrying an electric current in a magnetic field. Theplanar motor provides a single stage to replace conventional stackedstages, with the stage being electromagnetically suspended or levitatedfor enhanced performance and versatility.

[0003] Planar motors typically include a magnet array and a coil array.Several basic designs for planar motors are known, and are distinguishedbased on which of the components are positionally fixed and which movewith respect thereto. In a first design, commonly referred to as a“moving coil type” planar motor, the coil array moves with respect to apositionally fixed magnet array. In one embodiment, as disclosed in U.S.Pat. No. 6,097,114 to Hazelton and shown schematically in FIG. 1, amoving coil planar motor 100 includes a base 102 with a flat magnetarray 103 having a plurality of magnets 104. A single X coil 106 and twoY Coils 108, 110 are attached to the underside of a stage frame 112(drawn in dashed lines) suspended above and parallel to magnet array102. Y coils 108, 110 are similar in structure to one another and havecoil wires oriented to provide force substantially in a Y direction. Xcoil 106 and Y coils 108, 110 are similar in structure, but X coil 106has coil wires oriented to provide force substantially in an X directionperpendicular to the Y direction.

[0004] X coil 106 and Y coils 108, 110 permit movement of stage frame112. To provide force to stage frame 112 in the X direction relative tomagnet array 102, two phase, three phase, or multiphase commutatedelectric current is supplied to X coil 106 in a conventional manner by acommutation circuit and current source 114. To provide force to stageframe 112 in the Y direction, two phase, three phase, or multiphasecommutated electric current is supplied to either one or both of the Ycoils 108, 110 in a conventional manner by respective commutationcircuits and current sources 116 and/or 118. To provide rotationaltorque to frame 112 relative to magnet array 102 in a horizontal planeparallel to the X and Y axes, commutated electric current is supplied toeither of Y coils 108, 110 individually by respective commutationcircuits and current source 116 or 118. Alternatively, electric currentis supplied to both Y coils 108, 110 simultaneously but with oppositepolarities by respective commutation circuits and current sources 116,118, providing Y force to one of Y coils 108, 110 in one direction andthe other Y coil 108, 110 in an opposite direction, thereby generating atorque about an axis normal to the XY plane. This torque typicallycauses rotation of stage frame 112 in the XY plane.

[0005] In a second design, also disclosed in U.S. Pat. No. 6,097,114 toHazelton and shown schematically in FIG. 2, a “moving magnet type”planar motor includes a magnet array that moves with respect to apositionally fixed coil array. In one embodiment, moving magnet planarmotor 200 includes an upper surface of a flat base 202 that is coveredwith coil units 204. A positioning stage 206 is suspended above flatbase 202 and has an array of magnets 208 facing the upper surface offlat base 202. A conventional commutation circuit (not shown) controlsand supplies electric current to coil units 204 in accordance with thedesired direction of travel of positioning stage 206. Appropriatelycommutated electric current creates Lorentz forces, which propelpositioning stage 206 to a desired location, altitude, and attitude.

[0006] Suspension of a stage 112, 206 may be accomplished using avariety of techniques. For example, additional, permanent magnets may beprovided on the upper surface of a stage 112, 206 and on a stationaryframe located above the stage 112, 206 (not shown). Alternatively, anair bearing may be provided between a stage 112, 206 and its respectivebase 102, 202. Electromagnetic force generated by the motor may insteadprovide the necessary suspension force.

[0007] Despite these developments, there is a need for a planar motorcontrol that simultaneously controls translational forces in the X- andY-directions and θ_(z) rotational movement. In addition, in order toachieve smooth operation of planar motors, rigorous computational powermust be provided. For example, complex mathematical relationships mustbe evaluated to achieve the desired torque and translation in the X andY directions. To this end, significant CPU power typically is required.A need exists, therefore, for planar motor control using relationshipswith less complexity.

[0008] Also, there is a need for a planar motor control that permitstorque control with very low force ripple.

SUMMARY OF THE INVENTION

[0009] The present invention is related to a planar motor including acoil array having a plurality of coils, each coil fixed in position withrespect to the other coils, and a magnet array having a plurality ofmagnets, each magnet fixed in position with respect to the othermagnets, with the magnet array being movable above the coil array in atleast two degrees of translational freedom and at least one degree ofrotational freedom. The planar motor further includes a model-basedpredictive torque controller including a nonlinear current switchingmodel, with the torque controller configured to provide current toenergize each coil in response to the position of each magnet withrespect to a coil. The torque controller provides currents to the coilarray to at least substantially reduce force ripple during movement ofthe magnet array.

[0010] The torque controller may simultaneously stabilize translationaland rotational movement, and may compensate for torque produced bytranslation. The coil array may be square., and may include at least 25coils.

[0011] The present invention also is related to a method for controllinga planar motor for movement in three degrees of freedom. The methodincludes: positioning a movable magnet array over a fixed coil array,the coil array having coils generally disposed in a plane defining firstand second directions that are substantially orthogonal to one another,and the magnet array having magnets with magnetic fields; applyingcurrents to the coils following a nonlinear current switching model tocontrol movement of the magnet array and substantially reduce forceripple during the movement. The method may further include determining afirst translational force for the magnet array in the first directionand a second translational force for the magnet array in the seconddirection. In addition, the method may include determining a torque forthe magnet array in a third direction perpendicular to the first andsecond directions.

[0012] The present invention further is related to a planar motorincluding magnet array means, coil array means, and control meansproviding electric current to the coil array means for controlledmovement of the magnet array means in three degrees of freedom includingnon-linear current switching means for at least substantially reducingforce ripple during movement of the magnet array.

[0013] The present invention also is related to a stage system includinga planar motor. The planar motor includes: a coil array having aplurality of coils, each coil fixed in position with respect to theother coils; a magnet array having a plurality of magnets, each magnetfixed in position with respect to the other magnets, the magnet arraybeing movable above the coil array in at least two degrees oftranslational freedom and at least one degree of rotational freedom; anda model-based predictive torque controller comprising a nonlinearcurrent switching model, the torque controller configured to providecurrent to energize each coil in response to the position of each magnetwith respect to a coil. The torque controller provides currents to thecoil array to at least substantially reduce force ripple during movementof the magnet array.

[0014] Furthermore, the present invention is related to an exposureapparatus including an illumination system that supplies radiant energyand a stage system including a planar motor. The planar motor includes:a coil array having a plurality of coils, each coil fixed in positionwith respect to the other coils; a magnet array having a plurality ofmagnets, each magnet fixed in position with respect to the othermagnets, the magnet array being movable above the coil array in at leasttwo degrees of translational freedom and at least one degree ofrotational freedom; and a model-based predictive torque controllercomprising a nonlinear current switching model, the torque controllerconfigured to provide current to energize each coil in response to theposition of each magnet with respect to a coil. The torque controllerprovides currents to the coil array to at least substantially reduceforce ripple during movement of the magnet array, and the stage systemcarries at least one object disposed on a path of the radiant energy. Adevice can be manufactured with the exposure apparatus. Any of a varietyof devices such as semiconductor chips (e.g., integrated circuits orlarge-scale integrations), liquid crystal panels, CCDs, thin filmmagnetic heads, or micro-machines, can be manufactured with the exposureapparatus.

[0015] The present invention additionally is related to a waferincluding an image, wherein the image is formed with an exposureapparatus that includes an illumination system that supplies radiantenergy and a stage system that includes a planar motor. The planar motorincludes: a coil array having a plurality of coils, each coil fixed inposition with respect to the other coils; a magnet array having aplurality of magnets, each magnet fixed in position with respect to theother magnets, the magnet array being movable above the coil array in atleast two degrees of translational freedom and at least one degree ofrotational freedom; and a model-based predictive torque controllercomprising a nonlinear current switching model, the torque controllerconfigured to provide current to energize each coil in response to theposition of each magnet with respect to a coil. The torque controllerprovides currents to the coil array to at least substantially reduceforce ripple during movement of the magnet array, and the stage systemcarries at least one object disposed on a path of the radiant energy.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Preferred features of the present invention are disclosed in theaccompanying drawings, wherein similar reference characters denotesimilar elements throughout the several views, and wherein:

[0017]FIG. 1 is a perspective view schematically showing a prior artmoving coil planar motor;

[0018]FIG. 2 is a perspective view schematically showing a prior artmoving magnet planar motor;

[0019]FIG. 3 is a perspective view showing a moving magnet planar motoraccording to an embodiment of the present invention disposed at aninitial position with respect to the coil array;

[0020]FIG. 4 is a plan view of the magnet array of FIG. 3;

[0021]FIG. 5 is a plan view of the magnet array of FIG. 3 disposed abovea coil array, forming a planar motor;

[0022]FIG. 6 is a graph showing the magnet force constant of the planarmotor of FIG. 5;

[0023]FIG. 6A is a graphical representation of a moving magnet forceconstant coefficient;

[0024]FIG. 6B is a graphical representation of the Im x component of themagnetic force constant;

[0025]FIG. 6C is a graphical representation of the Im y component of themagnetic force constant;

[0026]FIG. 7 is a partial plan view of the planar motor of FIG. 5 withone row of magnets and one row of coils;

[0027]FIG. 8 is a plan view of the magnet array of FIG. 3 disposed atanother position with respect to the coil array;

[0028]FIG. 9 is an exemplar graph showing undesired torque behavior;

[0029]FIG. 9A is an exemplar graph related to torque compensation;

[0030]FIG. 9B is another exemplar graph related to torque compensation;

[0031]FIG. 10 is an exemplar graph showing torque compensation accordingto the present invention;

[0032]FIG. 11 is an exemplar graph showing translation compensationaccording to the present invention;

[0033]FIG. 12 is a block diagram of a position control system using anexemplary array of thirty-six coils in accordance with the presentinvention;

[0034]FIG. 13 is an elevational view, partially in section, showing amicrolithographic apparatus in accordance with the present invention;

[0035]FIG. 14 is a flowchart showing the fabrication of semiconductordevices; and

[0036]FIG. 15 is a flowchart showing details of the wafer processingstep of FIG. 14.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0037] Referring initially to FIG. 3, there is shown a perspective viewof a moving magnet embodiment of a planar motor 300 including such asquare flat planar coil array 302. Moving magnet planar motors suitablefor the present invention are disclosed, for example, in U.S. Pat. No.6,097,114 to Hazelton, U.S. Pat. No. 6,114,781 to Hazelton et al., andU.S. Pat. No. 6,188,147 B1 to Hazelton et al., the contents of which arehereby incorporated by reference in their entirety. A magnet array 304is attached to a moving portion of a positioning stage 306. Coils 308 ofcoil array 302 are attached to a fixed platen 310. In this embodiment,magnet array 304 is sized such that four groups of coils 308 (16 coils)fit underneath magnet array 304. Coils 308 can be switched electricallysuch that only the coils that are underneath magnet array 304 forproducing force are energized. The other coils are switched off tominimize heating of the system. Magnet array 304 is configured toprovide a magnetic flux field that interacts with coil array 302 toproduce forces to move positioning stage 306 in three degrees of freedom(conventionally designated X, Y, θ_(z)) above coil array 302. Althoughnot shown in FIG. 3, air bearings and associated smooth, hard surfacesmay be provided to facilitate movement of magnet array 304 with respectto coil array 302.

[0038] As shown in the plan view of FIG. 4, in the preferred embodiment,magnet array 304 includes centrally-located, full-sized square magnets312, peripherally-located half magnets 314, and quarter magnets 316 atthe four corners. Half magnets 314 generate substantially one-half ofthe magnetic flux of fall-sized magnets 312, while quarter magnets 316generate substantially one-quarter of said flux. The half magnets 314and quarter magnets 316 provide efficient magnetic flux coupling withfull-sized magnets 312. Magnet array 304 is disposed about a center ofgravity or origin 318, and magnets 312, 314, 316 form rows in the Xdirection and columns in the Y direction as defined by X and Ycoordinate axes. Using these axes, the magnetic pitch of the array isdefined as one-half the distance along a particular axis between centersof adjacent fall-sized magnets 312. Each full-sized magnet 312 has alength of about 1 pitch, p, and an area of about one pitch squared (p²),as shown graphically.

[0039] Turning to FIG. 5, magnet array 304 is disposed above coils 308which are arranged in the current embodiment in a 6×6 square array. Sixcoils are in each column C₀, C₁, C₂, C₃, C₄, C₅ and six coils are ineach row R₀, R₁, R₂, R₃, R₄, R₅, thus forming an array of thirty-sixcoils 308. As will be described shortly, the combination of magnet array304 and an array of coils 308 permits planar motor control in 3 degreesof freedom—x- and y-translation and z-rotation. Each coil 308 has alength of about 3 pitch, 3p, and an area of about 9p², as showngraphically. Persons of ordinary skill in the art will appreciate thatthe present invention may be readily adapted to control magnet arrays ofdifferent dimensions based on the teachings set forth herein.Preferably, a 5×5 or larger array of coils 308 is used, and the size ofthe coil array is selected in part based on the desired travel range formagnet array 304. In other embodiments, the numbers of rows and columnsin a magnet array may be substantially larger and/or the number of rowsand the number of columns may be unequal.

[0040]FIG. 6 shows the magnet force constant, K_(m), for magnet array304 of planar motor 300. In order to create X- and Y-translation forces,according to the Lorentz law, the two dimensional magnetic forceconstant, K_(m), may be mathematically derived for a 2-dimensionalplanar motor. The moving magnet force constant coefficient curve isshown in FIG. 6A. As can be seen from FIG. 6A, the force constantmagnitude has a trapezoid shape in both the x- and y-directions. Theforce constant coefficient curve may be used to derive equations for thex- and y-magnetic force constant. Referring to FIG. 6A, the movingmagnet force constant amplitude, A, in x-movement with respect toportion A₁ (corresponding to Im x) is as follows: $\begin{matrix}{{{A_{xx}\left( {x_{1},y_{1}} \right)}:=\left| \begin{matrix}\left. a\leftarrow{{1.0\quad {if}\quad 0} \leq {x_{1}} < 4.5} \right. \\\left. a\leftarrow{{{{- 1} \cdot \frac{1}{3} \cdot {x_{1}}} + {2.5\quad {if}\quad 4.5}} \leq {x_{1}} < 7.5} \right. \\\left. a\leftarrow{{0\quad {if}\quad 7.5} \leq {x_{1}}} \right.\end{matrix} \right.};} & (1) \\{{{A_{xy}\left( {x_{1},y_{1}} \right)}:=\left| \begin{matrix}\left. a\leftarrow{{1.0\quad {if}\quad 0} \leq {y_{1}} < 4.5} \right. \\\left. a\leftarrow{{{{- 0.5} \cdot {y_{1}}} + {3.25\quad {if}\quad 4.5}} \leq {y_{1}} < 5.5} \right. \\\left. a\leftarrow{{0.5\quad {if}\quad 5.5} \leq {y_{1}} < 6.5} \right. \\\left. a\leftarrow{{{{- 0.5} \cdot {y_{1}}} + {3.75\quad {if}\quad 6.5}} \leq {y_{1}} < 7.5} \right. \\\left. a\leftarrow{{0\quad {if}\quad 7.5} \leq {y_{1}}} \right.\end{matrix} \right.};} & (2)\end{matrix}$

A _(x)(x ₁ ,y ₁):=A _(xx)(x ₁ ,y ₁)·A _(xy)(x ₁ ,y ₁)·k _(x).  (3)

[0041] Further, with respect to portion A₂ the moving magnet forceconstant amplitude, A_(x) is as follows with respect to y-movement:$\begin{matrix}{{{A_{yx}\left( {x_{1},y_{1}} \right)}:=\left| \begin{matrix}\left. a\leftarrow{{{{- 0.5} \cdot {x_{1}}}\quad {if}\quad 0} \leq {x_{1}} < 4.5} \right. \\\left. a\leftarrow{{{{- 0.5} \cdot {x_{1}}} + {3.25\quad {if}\quad 4.5}} \leq {x_{1}} < 5.5} \right. \\\left. a\leftarrow{{0.5\quad {if}\quad 5.5} \leq {x_{1}} < 6.5} \right. \\\left. a\leftarrow{{{{- 0.5} \cdot {x_{1}}} + {3.75\quad {if}\quad 6.5}} \leq {x_{1}} < 7.5} \right. \\\left. a\leftarrow{{0\quad {if}\quad 7.5} \leq {x_{1}}} \right.\end{matrix} \right.};} & (4) \\{{{A_{yy}\left( {x_{1},y_{1}} \right)}:=\left| \begin{matrix}\left. a\leftarrow{{1.0\quad {if}\quad 0} \leq {y_{1}} < 4.5} \right. \\\left. a\leftarrow{{{{- 1} \cdot \frac{1}{3} \cdot {y_{1}}} + {2.5\quad {if}\quad 4.5}} \leq {y_{1}} < 7.5} \right. \\\left. a\leftarrow{{0\quad {if}\quad 7.5} \leq {y_{1}}} \right.\end{matrix} \right.};} & (5)\end{matrix}$

A _(y)(x ₁ ,y ₁):=A _(yx)(x ₁ ,y ₁)·A _(yy)(x ₁ ,y ₁)·k _(y).  (6)

[0042]FIGS. 6B and 6C show graphical representations of the Im x and Imy components, respectively, of the magnetic force constant, wherecoordinates (42, 42) are equivalent to position (0,0) at theintersection of the x- and y-axes in FIG. 5.

[0043] Referring again to FIG. 5, the magnet force constant for a givenrow of coils may be determined. For example, with the origin used for(x₁, y₁), the coils at positions (R₂, $\begin{matrix}{{k_{ma}\left( {x_{1},y_{1}} \right)}:=\begin{bmatrix}{{A_{x}\left( {{{- x_{1}} - 6},{- y_{1}}} \right)} \cdot P_{1}} & {{A_{x}\left( {{{- x_{1}} - 3},{- y_{1}}} \right)} \cdot P_{2}} & {{A_{x}\left( {{{- x_{1}} - 0},{- y_{1}}} \right)} \cdot P_{3}} \\{{A_{y}\left( {{{- x_{1}} - 6},{- y_{1}}} \right)} \cdot P_{4}} & {{A_{x}\left( {{{- x_{1}} - 3},{- y_{1}}} \right)} \cdot P_{5}} & {{A_{x}\left( {{{- x_{1}} - 0},{- y_{1}}} \right)} \cdot P_{6}} \\0 & 0 & 0\end{bmatrix}} & (7)\end{matrix}$

[0044] C₀), (R₂, C₁), and (R₂, C₂) contribute the following to themagnetic force constant:

[0045] where $\begin{matrix}{{{P_{1} = {{\sin \left( {{{- x_{1}} \cdot \frac{\pi}{2}} - {6 \cdot \frac{\pi}{2}}} \right)} \cdot {\cos \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{{P_{2} = {{\sin \left( {{{- x_{1}} \cdot \frac{\pi}{2}} - {3 \cdot \frac{\pi}{2}}} \right)} \cdot {\cos \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{{P_{3} = {{\sin \left( {{{- x_{1}} \cdot \frac{\pi}{2}} - {0 \cdot \frac{\pi}{2}}} \right)} \cdot {\cos \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{and}} & (8) \\{{{P_{4} = {{\cos \left( {{{- x_{1}} \cdot \frac{\pi}{2}} - {6 \cdot \frac{\pi}{2}}} \right)} \cdot {\sin \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{{P_{5} = {{\cos \left( {{{- x_{1}} \cdot \frac{\pi}{2}} - {3 \cdot \frac{\pi}{2}}} \right)} \cdot {\sin \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{P_{6} = {{\cos \left( {{{- x_{1}} \cdot \frac{\pi}{2}} - {0 \cdot \frac{\pi}{2}}} \right)} \cdot {{\sin \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}.}}}} & (9)\end{matrix}$

[0046] Similarly, the coils at positions (R₂, C₃), (R₂, C₄), and (R₂,C₅) contribute the following to the magnetic force constant:$\begin{matrix}{{{k_{mb}\left( {x_{1},y_{1}} \right)}:=\begin{bmatrix}{{A_{x}\left( {{{- x_{1}} + 3},y_{1}} \right)} \cdot P_{7}} & {{A_{x}\left( {{{- x_{1}} + 6},y_{1}} \right)} \cdot P_{8}} & {{A_{x}\left( {{{- x_{1}} + 9},y_{1}} \right)} \cdot P_{9}} \\{{A_{y}\left( {{{- x_{1}} + 3},y_{1}} \right)} \cdot P_{10}} & {{A_{x}\left( {{{- x_{1}} + 6},y_{1}} \right)} \cdot P_{11}} & {{A_{x}\left( {{{- x_{1}} + 9},y_{1}} \right)} \cdot P_{12}} \\0 & 0 & 0\end{bmatrix}}{where}} & (10) \\{{{P_{7} = {{\sin \left( {{{- x_{1}} \cdot \frac{\pi}{2}} + {3 \cdot \frac{\pi}{2}}} \right)} \cdot {\cos \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{{P_{8} = {{\sin \left( {{{- x_{1}} \cdot \frac{\pi}{2}} + {6 \cdot \frac{\pi}{2}}} \right)} \cdot {\cos \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{{P_{9} = {{\sin \left( {{{- x_{1}} \cdot \frac{\pi}{2}} + {9 \cdot \frac{\pi}{2}}} \right)} \cdot {\cos \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{and}} & (11) \\{{{P_{10} = {{\cos \left( {{{- x_{1}} \cdot \frac{\pi}{2}} + {3 \cdot \frac{\pi}{2}}} \right)} \cdot {\sin \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{{P_{11} = {{\cos \left( {{{- x_{1}} \cdot \frac{\pi}{2}} + {6 \cdot \frac{\pi}{2}}} \right)} \cdot {\sin \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}}};}{P_{12} = {{\cos \left( {{{- x_{1}} \cdot \frac{\pi}{2}} + {9 \cdot \frac{\pi}{2}}} \right)} \cdot {{\sin \left( {{- y_{1}} \cdot \frac{\pi}{2}} \right)}.}}}} & (12)\end{matrix}$

[0047] With reference to FIG. 7, row R₂ of coils 308 is shown withmagnets 312, 314 of magnet array 304 positioned thereabout. Magnets 312,314 span a total distance D, along the X-axis, which length is equal tothe length spanned by four complete coils 308. Thus, preferably, atleast five coils 308 are provided in each row so that magnets 312, 314may be translated with respect to coils 308. In addition, in thepreferred embodiment, distance D₁ is about 430 mm. The array of magnetsdisposed along the X-axis includes five full-sized magnets 312 and twohalf magnets 314, which is numerically equivalent to six full-sizedmagnets 312 each having an area of about one pitch squared (p²). Also,extending between half magnets 314 along the X-axis, open squarenon-magnet portions 320 use the equivalent area of six full-sizedmagnets 312, and so the magnet array uses the total equivalent area oftwelve full-sized magnets 312. The preferred embodiment has a pitch,therefore, of the ratio of D₁ to 12, or 430/12 mm. Notably, in theembodiment of planar motor 304 discussed herein, magnets 312, 314 alongthe X-axis alternate in North (N)-South (S) polarity. Each change inunit pitch also is equal to a 90° phase difference as encountered withsine or cosine functions; while magnet 312 at origin 318 is described ashaving an N-polarity and being at 0°, adjacent magnets with anS-polarity have a 180° phase difference.

[0048] The combination of row R₂ of coils 308 and magnets 312, 314 ofmagnet array 304 produces a translational force along the X-axis. Tocreate a four-phase linear motor, the magnet force constants, K_(m),located above each coil 308 are determined. The force constant of magnet312 located above coil 308 in column C₂, which coincides withcommutation origin 318, is constructed as follows:

K _(x) _(mag(0)) =K _(a) sin(x+0)  (13)

[0049] Thus, K_(a) is the peak-to-peak amplitude of the magnet forceconstant, K_(m). Similarly, the force constants of magnets 312 locatedabove coils 308 in columns C₃, C₂, respectively, are as follows:$\begin{matrix}{K_{x_{{mag}{(3)}}} = {K_{a}{\sin \left( {x + {3\frac{\pi}{2}}} \right)}}} & (14) \\{K_{x_{{mag}{({- 3})}}} = {K_{a}{\sin \left( {x - {3\frac{\pi}{2}}} \right)}}} & (15)\end{matrix}$

[0050] It should be noted that the factors of 3 in Eqs. 14 and 15 aboveare due to the offset of the respective coils a distance of 3 pitch fromorigin 318. Finally, the force constants of magnets 314 located abovecoils 308 in columns C₄, C₀, respectively, are as follows:$\begin{matrix}{{K_{x_{{mag}{({- 6})}}} = {\frac{1}{2}K_{a}{\sin \left( {x - {6\frac{\pi}{2}}} \right)}}};} & (16) \\{K_{x_{{mag}{(6)}}} = {\frac{1}{2}K_{a}{{\sin \left( {x + {6\frac{\pi}{2}}} \right)}.}}} & (17)\end{matrix}$

[0051] As indicated in Eqs. 16 and 17, the factors of 6 are due to theoffset of the respective coils a distance of 6 pitch from origin 318,while the factors of ½ account for the half-size of magnets 314.

[0052] Next, to create a driving force in the X-direction and located atthe center of each coil 308, assuming the current in each coil 308 hasthe same phase as the respective magnet force constant, the followingcurrents are required for coils 308 in columns C₁, C₂, C₃, C₄, C₅:$\begin{matrix}{{I_{x{({- 6})}} = {I\quad {\sin \left( {x - {6\frac{\pi}{2}}} \right)}}};} & (18) \\{{I_{x{({- 3})}} = {I\quad {\sin \left( {x - {3\frac{\pi}{2}}} \right)}}};} & (19) \\{{I_{x{(0)}} = {I\quad {\sin \left( {x + {0\frac{\pi}{2}}} \right)}}};} & (20) \\{{I_{x{(3)}} = {I\quad {\sin \left( {x + {3\frac{\pi}{2}}} \right)}}};} & (21) \\{{I_{x{(6)}} = {I\quad {\sin \left( {x + {6\frac{\pi}{2}}} \right)}}};} & (22)\end{matrix}$

[0053] Thus, a total driving force, F, is the summation of the productsof the individual driving forces and their respective force constants:$\begin{matrix}\begin{matrix}{F = \quad {{\frac{1}{2}I_{x{({- 6})}}K_{x_{{mag}{({- 6})}}}} + {I_{x{({- 3})}}K_{x_{{mag}{({- 3})}}}} + {I_{x{(0)}}K_{x_{{mag}{(0)}}}} +}} \\{\quad {{I_{x{(3)}}K_{x_{{mag}{(3)}}}} + {\frac{1}{2}I_{x{(6)}}K_{x_{{mag}{(6)}}}}}}\end{matrix} & (23)\end{matrix}$

[0054] Equation 24 may be simplified with the following relations.$\begin{matrix}{{{\frac{1}{2}I_{x{({- 6})}}K_{x_{{mag}{({- 6})}}}} = {{\frac{1}{2}I_{x{(6)}}K_{x_{{mag}{(6)}}}} = {\frac{1}{2}{IK}_{a}{\sin^{2}(x)}}}};} & (24)\end{matrix}$

I _(x(−3)) K _(x) _(mag(−3)) =I _(x(3)) K _(x) _(mag(3)) =IK _(a)cos²(x);  (25)

I _(x(0)K) _(xmag(0)) =IK _(a) sin²(x).  (26)

[0055] And upon simplification, the total force generated at row R₂becomes:

F=2IK _(a)[sin²(x)+cos²(x)]=2IK _(a)  (27)

[0056] Equation 27 may be further extended, so that the force generatedat row R₂ by coils 308 at locations (0,0), (3,0), (6,0), (−6,0), and(−3,0) is described as: Row 2: $\begin{matrix}\begin{matrix}{F_{x} = \quad {{K_{x}{\sin \left( {x + 0} \right)}{{\cos (y)}\left\lbrack {{I_{x}{\sin \left( {x + 0} \right)}{\cos (y)}} + {I_{y}{\sin (y)}{\cos \left( {x + 0} \right)}}} \right\rbrack}} +}} \\{\quad {{K_{x}{\sin \left( {x + 90} \right)}{{\cos (y)}\left\lbrack {{I_{x}{\sin \left( {x + 90} \right)}{\cos (y)}} + {I_{y}{\sin (y)}{\cos \left( {x + 90} \right)}}} \right\rbrack}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 180} \right)}{{\cos (y)}\left\lbrack {{I_{x}{\sin \left( {x + 180} \right)}{\cos (y)}} + {I_{y}{\sin (y)}{\cos \left( {x + 180} \right)}}} \right\rbrack}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 180} \right)}{{\cos (y)}\left\lbrack {{I_{x}{\sin \left( {x + 180} \right)}{\cos (y)}} + {I_{y}{\sin (y)}{\cos \left( {x + 180} \right)}}} \right\rbrack}} +}} \\{\quad {K_{x}{\sin \left( {x + 270} \right)}{{\cos (y)}\left\lbrack {{I_{x}{\sin \left( {x + 270} \right)}{\cos (y)}} + {I_{y}{\sin (y)}{\cos \left( {x + 270} \right)}}} \right\rbrack}}} \\{= \quad {2\quad I_{x}K_{a}{\cos^{2}(y)}}}\end{matrix} & (28)\end{matrix}$

[0057] As will be described shortly, if all coils 308 in rows R₀, R₁,R₂, R₃, R₄, R₅ are used to create a translational force in theX-direction, the equation for calculating the force simplifies to:

F_(x)=4I_(x)K_(a)   (29)

[0058] Similarly, the driving force in the Y-direction at column C₂created by coils 308 at locations (0,0), (0,3), (0,6), (0,−6), and(0,−3) is described as: Column 2: $\begin{matrix}\begin{matrix}{F_{y} = \quad {{K_{y}{\sin \left( {y + 0} \right)}{{\cos (x)}\left\lbrack {{I_{x}{\sin \left( {x + 0} \right)}{\cos (y)}} + {I_{y}{\sin (y)}{\cos \left( {x + 0} \right)}}} \right\rbrack}} +}} \\{\quad {{K_{y}{\sin \left( {y + 90} \right)}{{\cos (x)}\left\lbrack {{I_{x}{\sin \left( {x + 90} \right)}{\cos (y)}} + {I_{y}{\sin \left( {y + 90} \right)}{\cos (x)}}} \right\rbrack}} +}} \\{\quad {{\frac{1}{2}K_{y}{\sin \left( {y + 180} \right)}{{\cos (x)}\left\lbrack {{I_{x}{\sin \left( {x + 180} \right)}{\cos (y)}} + {I_{y}{\sin \left( {y + 180} \right)}{\cos (x)}}} \right\rbrack}} +}} \\{\quad {{\frac{1}{2}K_{y}{\sin \left( {y + 180} \right)}{{\cos (x)}\left\lbrack {{I_{x}{\sin \left( {x + 180} \right)}{\cos (y)}} + {I_{y}{\sin \left( {y + 180} \right)}{\cos (x)}}} \right\rbrack}} +}} \\{\quad {K_{y}{\sin \left( {y + 270} \right)}{{\cos (x)}\left\lbrack {{I_{x}{\sin \left( {x + 270} \right)}{\cos (y)}} + {I_{y}{\sin \left( {y + 270} \right)}{\cos (x)}}} \right\rbrack}}} \\{= \quad {2I_{y}K_{a}{\cos^{2}(x)}}}\end{matrix} & (30)\end{matrix}$

[0059] Accounting for the force contributed by all of coils 308 incolumns C₀, C₁, C₂, C₃, C₄, C₅, the total translational force in theY-direction simplifies to:

F_(y)=4I_(y)K_(a)   (31)

[0060] It is desirable to provide torque or yaw control (θ_(z)) for themoving magnet planar motor so that control of a third degree of freedomcomplements the X- and Y-direction translational force control alreadydiscussed. In order to calculate torque, the distance from each of thecoils to the center of gravity of the coil array must be known. To thisend, referring again to FIG. 5, it is noted that the distances betweeneach of coils 308 are fixed. Thus, as shown in FIG. 5, the center pointsCEN of coils 308 in rows R₀ and R₄ are offset in the y-direction bydistances L₀, L₄, respectively, from the x-axis that extends throughorigin 318, the center points CEN of coils 308 in rows R₁ and R₂ areoffset by distances L₁, L₃, respectively, and the center points CEN ofcoils 308 in row R₂ are offset by a distance L₂ from the X-axis, whichis zero. Rows R₀ and R₄ each produce one-half of the force produced byeach of rows R₁, R₂, R₃, because of the difference in size of themagnets in the rows. At the position shown in FIG. 5, magnet array 304is disposed only above rows R₀, R₁, R₂, R₃, R₄, so that the totaltranslation forces made by coils 308 in each of these rows is asfollows: $\begin{matrix}{{F_{row0} = \frac{2I_{u}K_{a}{\cos^{2}(y)}}{2}};} & (32)\end{matrix}$

 F _(row 1)=2I _(u) K _(a) sin²(y);  (33)

F _(row 2)=2(I _(u) +I _(l))K _(a) cos²(y);  (34)

F _(row 3)=2K _(j) K _(a) sin²(y);   (35) $\begin{matrix}{F_{row4} = {\frac{2I_{1}K_{a}{\cos^{2}(y)}}{2}.}} & (36)\end{matrix}$

[0061] where “upper” rows R₀, R₁, R₂ each have a current amplitude ofamps, and “lower” rows R₂, R₃, R₄ each have a current amplitude of I_(l)amps.

[0062] The torque equation thus is derived as follows: $\begin{matrix}\begin{matrix}{T_{{row}{(y)}} = \quad {{\frac{1}{2}{L_{0}(2)}I_{u}K_{a}{\cos^{2}(y)}} + {{L_{1}(2)}I_{u}K_{a}{\sin^{2}(y)}} +}} \\{\quad {{{L_{2}(2)}\left( {I_{u} + I_{1}} \right)K_{a}{\cos^{2}(y)}} -}} \\{\quad {{{L_{3}(2)}I_{1}K_{a}{\sin^{2}(y)}} - {\frac{1}{2}{L_{4}(2)}I_{1}K_{a}{\cos^{2}(y)}}}}\end{matrix} & (37)\end{matrix}$

[0063] The current used to generate the torque is defined such thatI_(u)=−I_(l), thus eliminating the third term of Eq. 37 which thenfurther simplifies to the following: $\begin{matrix}\begin{matrix}{T_{{row}{(y)}} = \quad {{\frac{1}{2}{L_{0}(2)}I_{u}K_{a}{\cos^{2}(y)}} + {{L_{1}(2)}I_{u}K_{a}{\sin^{2}(y)}} -}} \\{\quad {{{L_{3}(2)}I_{1}K_{a}{\sin^{2}(y)}} - {\frac{1}{2}{L_{4}(2)}I_{1}K_{a}{\cos^{2}(y)}}}} \\{= \quad {{\left( {L_{0} + L_{4}} \right)I_{u}K_{a}{\cos^{2}(y)}} + {\left( {L_{1} + L_{3}} \right)(2)I_{u}K_{a}{\sin^{2}(y)}}}}\end{matrix} & (38)\end{matrix}$

[0064] The offset distances L₀, L₁, L₂, L₃, L₄ are fixed, and in thepreferred embodiment, the value of (L₀+L₄) is fixed at 12 pitches whilethe value of (L₁+L₃) is fixed at 6 pitches. Equation 38 maybe furthersimplified as: $\begin{matrix}\begin{matrix}{T_{{row}{(y)}} = \quad {12I_{u}{K_{a}\left\lbrack {{\sin^{2}(y)} + {\cos^{2}(y)}} \right\rbrack}}} \\{= \quad {12K_{a}I_{u{({row})}}}}\end{matrix} & (39)\end{matrix}$

[0065] The θ_(z) torque thus may be created by a row of coils 308. Inaddition, a column of coils 308 also may produce the θ_(z) torque, whichis similarly determined to be as follows:

T_(column(x))=12K_(a)I_(u(column))  (40)

[0066] Moreover, both a row and column of coils 308 may be used toproduce the θ_(z) torque with one-half the desired torque produced byeach of the column and row.

[0067] In order to generate torque, the amplitude of the currentsupplied to a coil 308 is as follows: $\begin{matrix}{{I_{x} = \frac{F_{x}}{4K_{a}}};} & (41) \\{I_{y} = {\frac{F_{y}}{4K_{a}}.}} & (42)\end{matrix}$

[0068] The current for a row and column of coils is found to be asfollows: $\begin{matrix}{{I_{u{({row})}} = \frac{T_{{row}{(y)}}}{12K_{a}}};} & (43) \\{I_{u{({column})}} = {\frac{T_{{column}{(x)}}}{12K_{a}}.}} & (44)\end{matrix}$

[0069] With the above equations, the amplitude of the coil current maybe determined as a function of the desired translational force andtorque.

[0070] Equations 32 to 44 can be applied with both the X- and Y-axialpositions varying between about −0.5 pitch and about +0.5 pitch. Thisapplication constraint, however, places a significant restriction on theavailable planar motor moving area. To extend the area of movement, aswitching function is used for coil row energization, as follows:$\begin{matrix}{{I_{tzx}\left( {x_{1},{y_{1}T_{z_{1}}}} \right)}:=\left| \begin{matrix}\begin{matrix}\left. I_{{tzx}_{0}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{y} \cdot 2}} \right. & {{{if}\quad - 1.5} \leq y_{1} \leq 2.0} \\\left. I_{{tzx}_{0}}\leftarrow 0 \right. & {otherwise} \\\left. I_{{tzx}_{1}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{y} \cdot 2}} \right. & {{{if}\quad - 1.5} \leq y_{1} \leq 4.0} \\\left. I_{{tzx}_{1}}\leftarrow 0 \right. & {otherwise} \\\left. I_{{tzx}_{2}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{y} \cdot 2}} \right. & {{{if}\quad - 1.5} \leq y_{1} \leq 1.0} \\\left. I_{{tzx}_{2}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{y} \cdot 2}} \right. & {{{if}\quad 1.0} \leq y_{1} \leq 4.5} \\\left. I_{{tzx}_{2}}\leftarrow 0 \right. & {otherwise} \\\left. I_{{tzx}_{3}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{y} \cdot 2}} \right. & {{{if}\quad - 1.5} \leq y_{1} \leq 2.0} \\\left. I_{{tzx}_{3}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{y} \cdot 2}} \right. & {{{if}\quad 4.0} \leq y_{1} \leq 4.5} \\\left. I_{{tzx}_{3}}\leftarrow 0 \right. & {otherwise} \\\left. I_{{tzx}_{4}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{y} \cdot 2}} \right. & {{{if}\quad - 1.0} \leq y_{1} \leq 4.5} \\\left. I_{{tzx}_{4}}\leftarrow 0 \right. & {otherwise} \\\left. I_{{tzx}_{5}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{y} \cdot 2}} \right. & {{{if}\quad 1.0} \leq y_{1} \leq 4.5} \\\left. I_{{tzx}_{5}}\leftarrow 0 \right. & {otherwise}\end{matrix} \\\left. I_{tzx}\leftarrow\frac{I_{tzx}}{2} \right.\end{matrix} \right.} & (45)\end{matrix}$

[0071] A switching function for coil column energization is as follows:$\begin{matrix}{{I_{tzy}\left( {x_{1},y_{1},T_{z_{1}}} \right)}:={\begin{matrix}\left. I_{{tzy}_{0}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{x} \cdot 2}} \right. & \quad & {{{if}\quad - 1.5} \leq x_{1} \leq 2.0} \\\left. I_{{tzy}_{0}}\leftarrow 0 \right. & \quad & {otherwise} \\\left. I_{{tzy}_{1}}\leftarrow{- \frac{T_{z_{1}}}{6{k_{x} \cdot 2}}} \right. & \quad & {{{if}\quad - 1.5} \leq x_{1} \leq 4.0} \\\left. I_{{tzy}_{1}}\leftarrow 0 \right. & \quad & {otherwise} \\\left. I_{{tzy}_{2}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{x} \cdot 2}} \right. & \quad & {{{if}\quad - 1.5} \leq x_{1} \leq 1.0} \\\left. I_{{tzy}_{2}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{x} \cdot 2}} \right. & \quad & {{{if}\quad 1.0} \leq x_{1} \leq 4.5} \\\left. I_{{tzy}_{2}}\leftarrow 0 \right. & \quad & {otherwise} \\\left. I_{tzy3}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{x} \cdot 2}} \right. & \quad & {{{if}\quad - 1.5`} \leq x_{1} \leq 2.0} \\\left. I_{{tzy}_{3}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{x} \cdot 2}} \right. & \quad & {{{if}\quad 4.0} \leq x_{1} \leq 4.5} \\\left. I_{{tzy}_{3}}\leftarrow 0 \right. & \quad & {otherwise} \\\left. I_{{tzy}_{4}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{x} \cdot 2}} \right. & \quad & {{{if}\quad - 1.0} \leq x_{1} \leq 4.5} \\\left. I_{{tzy}_{4}}\leftarrow 0 \right. & \quad & {otherwise} \\\left. I_{{tzy}_{5}}\leftarrow{- \frac{T_{z_{1}}}{6 \cdot k_{x} \cdot 2}} \right. & \quad & {{{if}\quad 1.0} \leq x_{1} \leq 4.5} \\\left. I_{{tzy}_{5}}\leftarrow 0 \right. & \quad & {otherwise} \\\quad & \left. I_{tzy}\leftarrow\frac{I_{tzy}}{2} \right. & \quad\end{matrix}}} & (46)\end{matrix}$

[0072] For example, the switching function of Eq. 45 is applied asfollows. To create desired torque over the entire planar motor movingarea, a row R₀ activation function is used. When magnet array 304 islocated between −1.5 pitch and 2.0 pitch, row R₀ torque control currentis energized; otherwise, row R₀ torque control current is turned off.Similarly, row R₁ coils are energized when magnet array 304 is locatedbetween about −1.5 pitch to about 4.0 pitch; otherwise, row R₁ coils areturned off. In the preferred embodiment, the switching function of Eq.45 allows the desired torque to be generated with magnet array 304 at awide range of locations.

[0073] The simultaneous generation of X- and Y-translational movementand η_(z) rotational movement is governed by the following equations, inwhich the magnet force constant is multiplied by the sum of thecommutation functions for translational force and torque, for each rowof coils 308: $\begin{matrix}{\text{For row 2:}\begin{matrix}{F_{x} = \quad {{K_{x}{\sin \left( {x + 0} \right)}{{\cos (y)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r2})}}} \right){\sin \left( {x + 0} \right)}{\cos (y)}} +} \\{\left( {I_{y{({c2})}} + I_{t{({c2})}}} \right){\sin (y)}{\cos (x)}}\end{bmatrix}}} +}} \\{\quad {{K_{x}{\sin \left( {x + 90} \right)}{{\cos (y)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r2})}}} \right){\sin \left( {x + 90} \right)}{\cos (y)}} +} \\{\left( {I_{y{({c1})}} + I_{t{({c1})}}} \right){\sin (y)}{\cos \left( {x + 90} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 180} \right)}{{\cos (y)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r2})}}} \right){\sin \left( {x + 180} \right)}{\cos (y)}} +} \\{\left( {I_{y{({c0})}} + I_{t{({c0})}}} \right){\sin (y)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 0} \right)}{{\cos (y)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r2})}}} \right){\sin \left( {x + 0} \right)}{\cos (y)}} +} \\{\left( {I_{y{({c4})}} + I_{t{({c4})}}} \right){\sin (y)}{\cos (x)}}\end{bmatrix}}} +}} \\{\quad {K_{x}{\sin \left( {x + 270} \right)}{{\cos (y)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r2})}}} \right){\sin \left( {x + 270} \right)}{\cos (y)}} +} \\{\left( {I_{y{({c3})}} + I_{t{({c3})}}} \right){\sin (y)}{\cos \left( {x + 270} \right)}}\end{bmatrix}}}} \\{= \quad {2\left( {I_{x} + I_{t{({r2})}}} \right)K_{a}{\cos^{3}(y)}}} \\{= \quad {{2\quad I_{x}K_{a}{\cos^{2}(y)}} + {2I_{t{({r2})}}K_{a}{\cos^{2}(y)}}}}\end{matrix}} & (47) \\{\text{For row 1:}\begin{matrix}{F_{x} = \quad {{K_{x}{\sin \left( {x + 0} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r1})}}} \right){\sin \left( {x + 0} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c2})}} + I_{t{({c2})}}} \right){\sin \left( {y + 90} \right)}{\cos (x)}}\end{bmatrix}}} +}} \\{\quad {{K_{x}{\sin \left( {x + 90} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r1})}}} \right){\sin \left( {x + 90} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c1})}} + I_{t{({c1})}}} \right){\sin \left( {y + 90} \right)}{\cos \left( {x + 90} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 180} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r1})}}} \right){\sin \left( {x + 180} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c0})}} + I_{t{({c0})}}} \right){\sin \left( {y + 90} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 180} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r1})}}} \right){\sin \left( {x + 180} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c4})}} + I_{t{({c4})}}} \right){\sin \left( {y + 90} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {K_{x}{\sin \left( {x + 270} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r1})}}} \right){\sin \left( {x + 270} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c3})}} + I_{t{({c3})}}} \right){\sin \left( {y + 90} \right)}{\cos \left( {x + 270} \right)}}\end{bmatrix}}}} \\{= \quad {{2I_{x}K_{a}{\sin^{2}(y)}} + {2I_{t{({r1})}}K_{a}{\sin^{2}(y)}}}}\end{matrix}} & (48) \\{\text{For row 0:}\begin{matrix}{F_{x} = \quad {{\frac{1}{2}K_{x}{\sin \left( {x + 0} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r0})}}} \right){\sin \left( {x + 0} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c2})}} + I_{t{({c2})}}} \right){\sin \left( {y + 180} \right)}{\cos (x)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 90} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r0})}}} \right){\sin \left( {x + 90} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c1})}} + I_{t{({c1})}}} \right){\sin \left( {y + 180} \right)}{\cos \left( {x + 90} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{4}K_{x}{\sin \left( {x + 180} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r0})}}} \right){\sin \left( {x + 180} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c0})}} + I_{t{({c0})}}} \right){\sin \left( {y + 180} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{4}K_{x}{\sin \left( {x + 180} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r0})}}} \right){\sin \left( {x + 180} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c4})}} + I_{t{({c4})}}} \right){\sin \left( {y + 180} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {\frac{1}{2}K_{x}{\sin \left( {x + 270} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r0})}}} \right){\sin \left( {x + 270} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c3})}} + I_{t{({c3})}}} \right){\sin \left( {y + 180} \right)}{\cos \left( {x + 270} \right)}}\end{bmatrix}}}} \\{= \quad {{1I_{x}K_{a}{\cos^{2}(y)}} + {1I_{t{({r0})}}K_{a}{\cos^{2}(y)}}}}\end{matrix}} & (49) \\{\text{For row 3:}\begin{matrix}{F_{x} = \quad {{K_{x}{\sin \left( {x + 0} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r3})}}} \right){\sin \left( {x + 0} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c2})}} + I_{t{({c2})}}} \right){\sin \left( {y + 90} \right)}{\cos (x)}}\end{bmatrix}}} +}} \\{\quad {{K_{x}{\sin \left( {x + 90} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r3})}}} \right){\sin \left( {x + 90} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c1})}} + I_{t{({c1})}}} \right){\sin \left( {y + 90} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 180} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r3})}}} \right){\sin \left( {x + 180} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c0})}} + I_{t{({c0})}}} \right){\sin \left( {y + 90} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 180} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r3})}}} \right){\sin \left( {x + 180} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c4})}} + I_{t{({c4})}}} \right){\sin \left( {y + 90} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {{K_{x}{\sin \left( {x + 270} \right)}{{\cos \left( {y + 90} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r3})}}} \right){\sin \left( {x + 270} \right)}{\cos \left( {y + 90} \right)}} +} \\{\left( {I_{y{({c3})}} + I_{t{({c3})}}} \right){\sin \left( {y + 90} \right)}{\cos \left( {x + 270} \right)}}\end{bmatrix}}} +}} \\{= \quad {{2I_{x}K_{a}{\sin^{2}(y)}} + {2I_{t{({r3})}}K_{a}{\sin^{2}(y)}}}}\end{matrix}} & (50) \\{\text{For row 4:}\begin{matrix}{F_{x} = \quad {{\frac{1}{2}K_{x}{\sin \left( {x + 0} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r4})}}} \right){\sin \left( {x + 0} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c2})}} + I_{t{({c2})}}} \right){\sin \left( {y + 180} \right)}{\cos (x)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{2}K_{x}{\sin \left( {x + 90} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r4})}}} \right){\sin \left( {x + 90} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c1})}} + I_{t{({c1})}}} \right){\sin \left( {y + 180} \right)}{\cos \left( {x + 90} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{4}K_{x}{\sin \left( {x + 180} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r4})}}} \right){\sin \left( {x + 180} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c0})}} + I_{t{({c0})}}} \right){\sin \left( {y + 180} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {{\frac{1}{4}K_{x}{\sin \left( {x + 180} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r4})}}} \right){\sin \left( {x + 180} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c4})}} + I_{t{({c4})}}} \right){\sin \left( {y + 180} \right)}{\cos \left( {x + 180} \right)}}\end{bmatrix}}} +}} \\{\quad {\frac{1}{2}K_{x}{\sin \left( {x + 270} \right)}{{\cos \left( {y + 180} \right)}\begin{bmatrix}{{\left( {I_{x} + I_{t{({r4})}}} \right){\sin \left( {x + 270} \right)}{\cos \left( {y + 180} \right)}} +} \\{\left( {I_{y{({c3})}} + I_{t{({c3})}}} \right){\sin \left( {y + 180} \right)}{\cos \left( {x + 270} \right)}}\end{bmatrix}}}} \\{= \quad {{1I_{x}K_{a}{\cos^{2}(y)}} + {1I_{t{({r4})}}K_{a}{\cos^{2}(y)}}}}\end{matrix}} & (51)\end{matrix}$

[0074] The total translational force in the X-direction is thendetermined by summing the contributions from each row of coils 308:$\begin{matrix}\begin{matrix}{F_{x{({total})}} = \quad {F_{x{({r0})}} + F_{x{({r1})}} + F_{x{({r2})}} + F_{x{({r3})}} + F_{x{({r4})}}}} \\{= \quad {4I_{x}{K_{a}\left\lbrack {{\sin^{2}(y)} + {\cos^{2}(y)}} \right\rbrack}}} \\{= \quad {4K_{a}I_{x}}}\end{matrix} & (52)\end{matrix}$

[0075] Control of the θ_(z) rotational movement is accomplished byselecting the current I_(x) in Eqs. 47 to 51. A similar approach is usedto Y-direction translational control. Control of the rotational force isaccomplished by selecting the current I_(t(r)) in Eqs. 47 to 51. Torquecontrol currents are determined as follows, with magnet array 304located as shown in FIG. 5: $\begin{matrix}{I_{t_{r0}},{{I_{t_{r1}} = \frac{{Torque}_{x}}{12K_{a}}};}} & (53) \\{{I_{t_{r2}} = {{\frac{{Torque}_{x}}{12K_{a}} - \frac{{Torque}_{x}}{12K_{a}}} = 0}};} & (54) \\{I_{t_{r3}},{{I_{t_{r4}} = {- \frac{{Torque}_{x}}{12K_{a}}}};}} & (55) \\{\begin{matrix}{T_{x_{total}} = \quad {{6I_{t_{r1}}K_{a}{\sin^{2}(y)}} + {6I_{t_{r0}}K_{a}{\sin^{2}(y)}} + {6I_{t_{r4}}K_{a}{\cos^{2}(y)}} +}} \\{\quad {6I_{t_{r3}}K_{a}{\cos^{2}(y)}}} \\{= \quad {12I_{t_{r}}{K_{a}\left\lbrack {{\sin^{2}(y)} + {\cos^{2}(y)}} \right\rbrack}}} \\{= \quad {12\quad K_{a}I_{t_{r}}}}\end{matrix}{I_{t_{r}} = \frac{{Torque}_{x}}{12\quad K_{a}}}} & (56)\end{matrix}$

[0076] Turning now to the treatment of undesirable cross couplingbetween translational forces and θ_(z) rotational movement, it shouldfirst be noted that the symmetrical alignment of magnet array 304 withrespect to coils 308, as shown in FIG. 5, permits translation in theX-direction without undesired torque. However, if the position of magnetarray 304 with respect to coils 308 is changed, for example, to thatshown in FIG. 8, driving translational forces are no longersymmetrically generated and torque compensation is desired. Suchcompensation may be achieved with the following method: (1) applying anX-direction translational force to magnet array 304 and measuring theundesired torque; (2) normalizing the measured undesirable torque tocreate a predictor or model of the behavior with the undesired torqueoutput preferably determined as a function of movement or offset ofmagnet array 304 in the Y-direction, measured in pitch; (3)substantially canceling the undesired torque using the behavior modeland the torque control of Eqs. 52 to 56.

[0077] An exemplar model of undesirable torque behavior is shown in FIG.9, with undesired torque (measured in units of Newton-meters) graphed asa function of displacement (measured in units of pitch). For example,movement of +1 pitch results in undesired torque of about 50 N-pitch.

[0078] Further, an exemplar model of undesired torque compensation forF_(x) is shown in FIG. 9A and a compensation model for F_(y) is shown inFIG. 9B.

[0079] Referring to FIGS. 10 and 11, torque and translation forceoutputs are shown in response to translation force compensation of 100N-pitch and 100 N, respectively, as magnet array 304 moves from y=−1.5pitch to y=4.5 pitch. Coupling between translational force control andθ_(z) rotational movement control is shown to be substantially withoutcoupling, with X-translation and θ_(z) rotation being independent.

[0080]FIG. 12 is a block diagram of a position control system 350 usingan exemplary array of thirty-six coils 308 according to the presentinvention. Blocks B₁ and B₂, for example, represent the undesired torquecompensation maps shown graphically in FIGS. 9A and 9B. The switchfunction of block B₃ may be the switch function of Equations 44 and 45as described above. The commutation block, B₄, may be governed byEquations 47 to 51, particularly the portion in square brackets. Thethirty-six coil current output of amplifier block 352 is supplied to aposition loop control at block B₅, at which time the force constant ismultiplied by the above-mentioned bracketed commutation current inEquations 47 to 51. A total of twenty-five portions in square bracketsare found in these equations, instead of 36, because the coil current iszero to position (R₅, C₅). Outputs x and y are measured in pitch, whileoutput θ is measured in radians. Element 354 represents the amplitude ofthe control current for rows R₀ to R₅, while element 356 represents theamplitude of the control current for columns C₀ to C₅.

[0081]FIG. 13 is an elevational view, partially in section, showing amicrolithographic apparatus 400 incorporating a planar motor-drivenpositioning stage 402 in accordance with the present invention.Microlithographic apparatus 400, such as described in U.S. Pat. No.5,528,118 to Lee, includes an upper optical system 404 and a lower wafersupport and positioning system 406. Optical system 404 includes anilluminator 408 containing a lamp LMP, such as a mercury vapor lamp, andan ellipsoidal mirror EM surrounding lamp LMP. Illuminator 408 alsocomprises an optical integrator, such as a fly's eye lens FEL, producingsecondary light source images, and a condenser lens CL for illuminatinga reticle (mask) R with uniform light flux. A mask holder RST holdingmask or reticle R is mounted above a lens barrel PL of a projectionoptical system. A lens barrel PL is fixed on a part of a column assembly410 which is supported on a plurality of rigid arms 412, each mounted onthe top portion of an isolation pad or block system 414.Microlithographic apparatus 400 exposes a pattern of the reticle R ontoa wafer W, while mask holder RST and positioning stage 402 are movingsynchronously relative to illuminator 408.

[0082] Inertial or seismic blocks 416 are located on the system, e.g.mounted on arms 412. Blocks 416 can take the form of a cast box whichcan be filled with sand at the operation site to reduce the shippingweight of apparatus 400. An object or positioning stage base 418 issupported from arms 412 by depending blocks 416 and depending bars 420and horizontal bars 422. Positioning stage 402 carrying wafer W issupported in a movable fashion by positioning stage base 418. A reactionframe 424 carries a magnet array (not shown) and drives positioningstage 402 in cooperation with a moving coil array (not shown). Reactionframe 424 is isolated from positioning stage base 418 in terms ofvibration relative to a foundation 426, when a force is generated aspositioning stage 402 is driven. Positioning stage 402 and/or maskholder RST can be driven by a planar motor such as planar motor 300described above.

[0083] There are a number of different types of photolithographicdevices. For example, exposure apparatus 400 can be used as a scanningtype photolithography system which exposes the pattern from reticle Ronto wafer W with reticle R and wafer W moving synchronously. In ascanning type lithographic device, reticle R is moved perpendicular toan optical axis of lens assembly 404 by reticle stage RST and wafer W ismoved perpendicular to an optical axis of lens assembly 404 by waferstage 402. Scanning of reticle R and wafer W occurs while reticle R andwafer W are moving synchronously.

[0084] Alternately, exposure apparatus 400 can be a step-and-repeat typephotolithography system that exposes reticle R while reticle R and waferW are stationary. In the step and repeat process, wafer W is in aconstant position relative to reticle R and lens assembly 404 during theexposure of an individual field. Subsequently, between consecutiveexposure steps, wafer W is consecutively moved by wafer stage 402perpendicular to the optical axis of lens assembly 404 so that the nextfield of semiconductor wafer W is brought into position relative to lensassembly 404 and reticle R for exposure. Following this process, theimages on reticle R are sequentially exposed onto the fields of wafer Wso that the next field of semiconductor wafer W is brought into positionrelative to lens assembly 404 and reticle R.

[0085] However, the use of exposure apparatus 400 provided herein is notlimited to a photolithography system for semiconductor manufacturing.Exposure apparatus 400, for example, can be used as an LCDphotolithography system that exposes a liquid crystal display devicepattern onto a rectangular glass plate or a photolithography system formanufacturing a thin film magnetic head. Further, the present inventioncan also be applied to a proximity photolithography system that exposesa mask pattern by closely locating a mask and a substrate without theuse of a lens assembly. Additionally, the present invention providedherein can be used in other devices, including other semiconductorprocessing equipment, machine tools, metal cutting machines, andinspection machines.

[0086] The illumination source 408 can be g-line (436 nm), i-line (365nm), KrF excimer laser (248 nm), ArF excimer laser (193 nm) and F₂ laser(157 nm). Alternatively, illumination source 408 can also use chargedparticle beams such as x-ray and electron beams. For instance, in thecase where an electron beam is used, thermionic emission type lanthanumhexaboride (LaB₆) or tantalum (Ta) can be used as an electron gun.Furthermore, in the case where an electron beam is used, the structurecould be such that either a mask is used or a pattern can be directlyformed on a substrate without the use of a mask.

[0087] With respect to lens assembly 404, when far ultra-violet rayssuch as the excimer laser are used, glass materials such as quartz andfluorite that transmit far ultra-violet rays are preferably used. Whenthe F₂ type laser or x-ray is used, lens assembly 404 should preferablybe either catadioptric or refractive (a reticle should also preferablybe a reflective type), and when an electron beam is used, electronoptics should preferably comprise electron lenses and deflectors. Theoptical path for the electron beams should be in a vacuum.

[0088] Also, with an exposure device that employs vacuum ultra-violetradiation (VUV of wavelength 200 nm or lower, use of the catadioptrictype optical system can be considered. Examples of the catadioptric typeof optical system include the disclosure Japan Patent ApplicationDisclosure No. 8-171054 published in the Official Gazette for Laid-OpenPatent Applications and its counterpart U.S. Pat. No. 5,668,672, as wellas Japan Patent Application Disclosure No. 10-20195 and its counterpartU.S. Pat. No. 5,835,275. In these cases, the reflecting optical devicecan be a catadioptric optical system incorporating a beam splitter andconcave mirror. Japan Patent Application Disclosure No. 8-334695published in the Official Gazette for Laid-Open Patent Applications andits counterpart U.S. Pat. No. 5,689,377 as well as Japan PatentApplication Disclosure No. 10-3039 and its counterpart European PatentApplication EP 0816892 A2 also use a reflecting-refracting type ofoptical system incorporating a concave mirror, etc., but without a beamsplitter, and can also be employed with this invention. The disclosuresin the above-mentioned U.S. patents, European patent application, aswell as the Japan patent applications published in the Official Gazettefor Laid-Open Patent Applications are incorporated herein by reference.

[0089] Further, in photolithography systems, when linear motors (seeU.S. Pat. Nos. 5,623,853 or 5,528,118) are used in a wafer stage or areticle stage, the linear motors can be either an air levitation typeemploying air bearings or a magnetic levitation type using Lorentz forceor reactance force. Additionally, the stage could move along a guide, orit could be a guideless type stage which uses no guide. The disclosuresin U.S. Pat. Nos. 5,623,853 and 5,528,118 are incorporated herein byreference.

[0090] Alternatively, one of the stages could be driven by a planarmotor, which drives the stage by electromagnetic force generated by amagnet unit having two-dimensionally arranged magnets and an armaturecoil unit having two-dimensionally arranged coils in facing positions.With this type of driving system, either one of the magnet unit or thearmature coil unit is connected to the stage and the other unit ismounted on the moving plane side of the stage.

[0091] Movement of the stages as described above generates reactionforces which can affect performance of the photolithography system.Reaction forces generated by the wafer (substrate) stage motion can bemechanically released to the floor (ground) by use of a frame member asdescribed in U.S. Pat. No. 5,528,118 and published Japanese PatentApplication Disclosure No. 8-166475. Additionally, reaction forcesgenerated by the reticle (mask) stage motion can be mechanicallyreleased to the floor (ground) by use of a frame member as described inU.S. Pat. No. 5,874,820 and published Japanese Patent ApplicationDisclosure No. 8-330224. The disclosures in U.S. Pat. Nos. 5,528,118 and5,874,820 and Japanese Patent Application Disclosure No. 8-330224 areincorporated herein by reference.

[0092] As described above, a photolithography system according to theabove-described embodiments can be built by assembling varioussubsystems, including each element listed in the appended claims, insuch a manner that prescribed mechanical accuracy, electrical accuracyand optical accuracy are maintained. In order to maintain the variousaccuracies, prior to and following assembly, every optical system isadjusted to achieve its optical accuracy. Similarly, every mechanicalsystem and every electrical system are adjusted to achieve theirrespective mechanical and electrical accuracies. The process ofassembling each subsystem into a photolithography system includesmechanical interfaces, electrical circuit wiring connections and airpressure plumbing connections between each subsystem. Needless to say,there is also a process where each subsystem is assembled prior toassembling a photolithography system from the various subsystems. Once aphotolithography system is assembled using the various subsystems, totaladjustment is performed to make sure that every accuracy is maintainedin the complete photolithography system. Additionally, it is desirableto manufacture an exposure system in a clean room where the temperatureand humidity are controlled.

[0093] Further, semiconductor devices can be fabricated using theabove-described systems, by the process shown generally in FIG. 14. Instep 501 the device's function and performance characteristics aredesigned. Next, in step 502, a mask (reticle) having a pattern isdesigned according to the previous designing step, and in a parallelstep 503, a wafer is made from a silicon material. The mask patterndesigned in step 502 is exposed onto the wafer from step 503 in step 504by a photolithography system described hereinabove consistent with theprinciples of the present invention. In step 505 the semiconductordevice is assembled (including the dicing process, bonding process andpackaging process), and then finally the device is inspected in step506.

[0094]FIG. 15 illustrates a detailed flowchart example of theabove-mentioned step 504 in the case of fabricating semiconductordevices. In step 511 (oxidation step), the wafer surface is oxidized. Instep 512 (CVD step), an insulation film is formed on the wafer surface.In step 513 (electrode formation step), electrodes are formed on thewafer by vapor deposition. In step 514 (ion implantation step), ions areimplanted in the wafer. The above-mentioned steps 511-514 form thepreprocessing steps for wafers during wafer processing, and selection ismade at each step according to processing requirements.

[0095] At each stage of wafer processing, when the above-mentionedpreprocessing steps have been completed, the following post-processingsteps are implemented. During post-processing, initially, in step 515(photoresist formation step), photoresist is applied to a wafer. Next,in step 516 (exposure step), the above-mentioned exposure device is usedto transfer the circuit pattern of a mask (reticle) to a wafer. Then, instep 517 (developing step), the exposed wafer is developed, and in step518 (etching step), parts other than residual photoresist (exposedmaterial surface) are removed by etching. In step 519 (photoresistremoval step), unnecessary photoresist remaining after etching isremoved.

[0096] Multiple circuit patterns are formed by repetition of thesepreprocessing and post-processing steps.

[0097] It will be apparent to those skilled In the art that variousmodifications and variations can be made in the methods described, inthe stage device, the control system, the material chosen for thepresent invention, and in construction of the photolithography systemsas well as other aspects of the invention without departing from thescope or spirit of the invention.

[0098] While various descriptions of the present invention are describedabove, it should be understood that the various features can be usedsingly or in any combination thereof. Therefore, this invention is notto be limited to only the specifically preferred embodiments depictedherein.

[0099] Further, it should be understood that variations andmodifications within the spirit and scope of the invention may occur tothose skilled in the art to which the invention pertains. For example,magnet arrays and coil arrays having a different number of magnetsand/or coils, respectively, from those discussed in detail herein may beused in accordance with the principles of the present invention.Accordingly, all expedient modifications readily attainable by oneversed in the art from the disclosure set forth herein that are withinthe scope and spirit of the present invention are to be included asfurther embodiments of the present invention. The scope of the presentinvention is accordingly defined as set forth in the appended claims.

What is claimed is:
 1. A planar motor comprising: a coil array having aplurality of coils, each coil fixed in position with respect to theother coils; a magnet array having a plurality of magnets, each magnetfixed in position with respect to the other magnets, the magnet arraybeing movable above the coil array in at least two degrees oftranslational freedom and at least one degree of rotational freedom; anda model-based predictive torque controller comprising a nonlinearcurrent switching model, the torque controller configured to providecurrent to energize each coil in response to the position of each magnetwith respect to a coil; wherein the torque controller provides currentsto the coil array to at least substantially reduce force ripple duringmovement of the magnet array.
 2. The planar motor of claim 1, whereinthe torque controller simultaneously stabilizes translational androtational movement.
 3. The planar motor of claim 1, wherein the torquecontroller compensates for torque produced by translation.
 4. The planarmotor of claim 1, wherein the coil array is square.
 5. The planar motorof claim 4, wherein the coil array comprises at least 25 coils.
 6. Amethod for controlling a planar motor for movement in three degrees offreedom, the method comprising: positioning a movable magnet array overa fixed coil array, said coil array having coils generally disposed in aplane defining first and second directions that are substantiallyorthogonal to one another, and said magnet array having magnets withmagnetic fields; applying currents to said coils following a nonlinearcurrent switching model to control movement of said magnet array andsubstantially reduce force ripple during said movement.
 7. The method ofclaim 6, further comprising: determining a first translational force forsaid magnet array in said first direction and a second translationalforce for said magnet array in said second direction.
 8. The method ofclaim 6, further comprising: determining a torque for said magnet arrayin a third direction perpendicular to said first and second directions.9. A planar motor comprising: magnet array means; coil array means; andcontrol means providing electric current to said coil array means forcontrolled movement of said magnet array means in three degrees offreedom including non-linear current switching means for at leastsubstantially reducing force ripple during movement of said magnetarray.
 10. A stage system comprising a planar motor, said planar motorcomprising: a coil array having a plurality of coils, each coil fixed inposition with respect to the other coils; a magnet array having aplurality of magnets, each magnet fixed in position with respect to theother magnets, the magnet array being movable above the coil array in atleast two degrees of translational freedom and at least one degree ofrotational freedom; and a model-based predictive torque controllercomprising a nonlinear current switching model, the torque controllerconfigured to provide current to energize each coil in response to theposition of each magnet with respect to a coil; wherein the torquecontroller provides currents to the coil array to at least substantiallyreduce force ripple during movement of the magnet array.
 11. An exposureapparatus comprising an illumination system that supplies radiant energyand a stage system comprising a planar motor, the planar motorcomprising: a coil array having a plurality of coils, each coil fixed inposition with respect to the other coils; a magnet array having aplurality of magnets, each magnet fixed in position with respect to theother magnets, the magnet array being movable above the coil array in atleast two degrees of translational freedom and at least one degree ofrotational freedom; and a model-based predictive torque controllercomprising a nonlinear current switching model, the torque controllerconfigured to provide current to energize each coil in response to theposition of each magnet with respect to a coil; wherein the torquecontroller provides currents to the coil array to at least substantiallyreduce force ripple during movement of the magnet array, and wherein thestage system carries at least one object disposed on a path of theradiant energy.
 12. A device manufactured with the exposure apparatus ofclaim
 11. 13. A wafer comprising an image, wherein said image is formedwith an exposure apparatus comprising an illumination system thatsupplies radiant energy and a stage system comprising a planar motor,the planar motor comprising: a coil array having a plurality of coils,each coil fixed in position with respect to the other coils; a magnetarray having a plurality of magnets, each magnet fixed in position withrespect to the other magnets, the magnet array being movable above thecoil array in at least two degrees of translational freedom and at leastone degree of rotational freedom; and a model-based predictive torquecontroller comprising a nonlinear current switching model, the torquecontroller configured to provide current to energize each coil inresponse to the position of each magnet with respect to a coil; whereinthe torque controller provides currents to the coil array to at leastsubstantially reduce force ripple during movement of the magnet array,and wherein the stage system carries at least one object disposed on apath of the radiant energy.